(Non-)Linear Tests for Null Hypotheses, Equivalence, Non Superiority, and Non Inferiority

Uncertainty estimates are calculated as first-order approximate standard errors for linear or non-linear functions of a vector of random variables with known or estimated covariance matrix. In that sense, hypotheses emulates the behavior of the excellent and well-established car::deltaMethod and car::linearHypothesis functions, but it supports more models; requires fewer dependencies; expands the range of tests to equivalence and superiority/inferiority; and offers convenience features like robust standard errors.

To learn more, read the hypothesis tests vignette, visit the package website, or scroll down this page for a full list of vignettes:

Warning #1: Tests are conducted directly on the scale defined by the type argument. For some models, it can make sense to conduct hypothesis or equivalence tests on the "link" scale instead of the "response" scale which is often the default.

Warning #2: For hypothesis tests on objects produced by the marginaleffects package, it is safer to use the hypothesis argument of the original function. Using hypotheses() may not work in certain environments, in lists, or when working programmatically with *apply style functions.

Usage

hypotheses(
  model,
  hypothesis = NULL,
  vcov = NULL,
  conf_level = 0.95,
  df = Inf,
  equivalence = NULL,
  FUN = NULL,
  ...
)

Arguments

model

Model object or object generated by the comparisons(), slopes(), predictions(), or marginal_means() functions.

hypothesis

specify a hypothesis test or custom contrast using a numeric value, vector, or matrix, a string, or a string formula.

Numeric:
- Single value: the null hypothesis used in the computation of Z and p (before applying transform).
- Vector: Weights to compute a linear combination of (custom contrast between) estimates. Length equal to the number of rows generated by the same function call, but without the hypothesis argument.
- Matrix: Each column is a vector of weights, as describe above, used to compute a distinct linear combination of (contrast between) estimates. The column names of the matrix are used as labels in the output.
String formula to specify linear or non-linear hypothesis tests. If the term column uniquely identifies rows, terms can be used in the formula. Otherwise, use b1, b2, etc. to identify the position of each parameter. Examples:
- hp = drat
- hp + drat = 12
- b1 + b2 + b3 = 0
String:
- "pairwise": pairwise differences between estimates in each row.
- "reference": differences between the estimates in each row and the estimate in the first row.
- "sequential": difference between an estimate and the estimate in the next row.
- "revpairwise", "revreference", "revsequential": inverse of the corresponding hypotheses, as described above.
See the Examples section below and the vignette: https://vincentarelbundock.github.io/marginaleffects/articles/hypothesis.html

vcov

Type of uncertainty estimates to report (e.g., for robust standard errors). Acceptable values:

FALSE: Do not compute standard errors. This can speed up computation considerably.
TRUE: Unit-level standard errors using the default vcov(model) variance-covariance matrix.
String which indicates the kind of uncertainty estimates to return.
- Heteroskedasticity-consistent: "HC", "HC0", "HC1", "HC2", "HC3", "HC4", "HC4m", "HC5". See ?sandwich::vcovHC
- Heteroskedasticity and autocorrelation consistent: "HAC"
- Mixed-Models degrees of freedom: "satterthwaite", "kenward-roger"
- Other: "NeweyWest", "KernHAC", "OPG". See the sandwich package documentation.
One-sided formula which indicates the name of cluster variables (e.g., ~unit_id). This formula is passed to the cluster argument of the sandwich::vcovCL function.
Square covariance matrix
Function which returns a covariance matrix (e.g., stats::vcov(model))

conf_level

numeric value between 0 and 1. Confidence level to use to build a confidence interval.

df

Degrees of freedom used to compute p values and confidence intervals. A single numeric value between 1 and Inf. When df is Inf, the normal distribution is used. When df is finite, the t distribution is used. See insight::get_df for a convenient function to extract degrees of freedom. Ex: slopes(model, df = insight::get_df(model))

equivalence

Numeric vector of length 2: bounds used for the two-one-sided test (TOST) of equivalence, and for the non-inferiority and non-superiority tests. See Details section below.

FUN

NULL or function.

NULL (default): hypothesis test on a model's coefficients, or on the quantities estimated by one of the marginaleffects package functions.
Function which accepts a model object and returns a numeric vector or a data.frame with two columns called term and estimate. This argument can be useful when users want to conduct a hypothesis test on an arbitrary function of quantities held in a model object.

...

Additional arguments are passed to the predict() method supplied by the modeling package.These arguments are particularly useful for mixed-effects or bayesian models (see the online vignettes on the marginaleffects website). Available arguments can vary from model to model, depending on the range of supported arguments by each modeling package. See the "Model-Specific Arguments" section of the ?marginaleffects documentation for a non-exhaustive list of available arguments.

Equivalence, Inferiority, Superiority

\(\theta\) is an estimate, \(\sigma_\theta\) its estimated standard error, and \([a, b]\) are the bounds of the interval supplied to the equivalence argument.

Non-inferiority:

\(H_0\): \(\theta \leq a\)
\(H_1\): \(\theta > a\)
\(t=(\theta - a)/\sigma_\theta\)
p: Upper-tail probability

Non-superiority:

\(H_0\): \(\theta \geq b\)
\(H_1\): \(\theta < b\)
\(t=(\theta - b)/\sigma_\theta\)
p: Lower-tail probability

Equivalence: Two One-Sided Tests (TOST)

p: Maximum of the non-inferiority and non-superiority p values.

Thanks to Russell V. Lenth for the excellent emmeans package and documentation which inspired this feature.

Examples

library(marginaleffects)
mod <- lm(mpg ~ hp + wt + factor(cyl), data = mtcars)

# When `FUN` and `hypotheses` are `NULL`, `hypotheses()` returns a data.frame of parameters
hypotheses(mod)
#> 
#>  Term Estimate Std. Error     z Pr(>|z|)   2.5 %    97.5 %
#>    b1  35.8460      2.041 17.56   <0.001 31.8457 39.846319
#>    b2  -0.0231      0.012 -1.93   0.0531 -0.0465  0.000306
#>    b3  -3.1814      0.720 -4.42   <0.001 -4.5918 -1.771012
#>    b4  -3.3590      1.402 -2.40   0.0166 -6.1062 -0.611803
#>    b5  -3.1859      2.170 -1.47   0.1422 -7.4399  1.068169
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

# Test of equality between coefficients
hypotheses(mod, hypothesis = "hp = wt")
#> 
#>     Term Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>  hp = wt     3.16       0.72 4.39   <0.001  1.75   4.57
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

# Non-linear function
hypotheses(mod, hypothesis = "exp(hp + wt) = 0.1")
#> 
#>                Term Estimate Std. Error     z Pr(>|z|)  2.5 %  97.5 %
#>  exp(hp + wt) = 0.1  -0.0594     0.0292 -2.04   0.0418 -0.117 -0.0022
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

# Robust standard errors
hypotheses(mod, hypothesis = "hp = wt", vcov = "HC3")
#> 
#>     Term Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>  hp = wt     3.16      0.805 3.92   <0.001  1.58   4.74
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

# b1, b2, ... shortcuts can be used to identify the position of the
# parameters of interest in the output of FUN
hypotheses(mod, hypothesis = "b2 = b3")
#> 
#>     Term Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>  b2 = b3     3.16       0.72 4.39   <0.001  1.75   4.57
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

# term names with special characters have to be enclosed in backticks
hypotheses(mod, hypothesis = "`factor(cyl)6` = `factor(cyl)8`")
#> 
#>                             Term Estimate Std. Error      z Pr(>|z|) 2.5 %
#>  `factor(cyl)6` = `factor(cyl)8`   -0.173       1.65 -0.105    0.917 -3.41
#>  97.5 %
#>    3.07
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

mod2 <- lm(mpg ~ hp * drat, data = mtcars)
hypotheses(mod2, hypothesis = "`hp:drat` = drat")
#> 
#>              Term Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %
#>  `hp:drat` = drat    -6.08       2.89 -2.1   0.0357 -11.8 -0.405
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

# predictions(), comparisons(), and slopes()
mod <- glm(am ~ hp + mpg, data = mtcars, family = binomial)
cmp <- comparisons(mod, newdata = "mean")
hypotheses(cmp, hypothesis = "b1 = b2")
#> 
#>   Term Estimate Std. Error     z Pr(>|z|)  2.5 %  97.5 %
#>  b1=b2   -0.288      0.125 -2.31   0.0209 -0.532 -0.0435
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

mfx <- slopes(mod, newdata = "mean")
hypotheses(cmp, hypothesis = "b2 = 0.2")
#> 
#>    Term Estimate Std. Error     z Pr(>|z|)  2.5 % 97.5 %
#>  b2=0.2    0.101      0.131 0.774    0.439 -0.155  0.358
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

pre <- predictions(mod, newdata = datagrid(hp = 110, mpg = c(30, 35)))
hypotheses(pre, hypothesis = "b1 = b2")
#> 
#>   Term Estimate Pr(>|z|)    2.5 % 97.5 %
#>  b1=b2  0.00184   0.0264 7.07e-06  0.324
#> 
#> Columns: term, estimate, p.value, conf.low, conf.high 
#> 

# The `FUN` argument can be used to compute standard errors for fitted values
mod <- glm(am ~ hp + mpg, data = mtcars, family = binomial)

f <- function(x) predict(x, type = "link", newdata = mtcars)
p <- hypotheses(mod, FUN = f)
head(p)
#> 
#>  Term Estimate Std. Error      z Pr(>|z|) 2.5 % 97.5 %
#>    b1   -1.098      0.716 -1.534    0.125 -2.50  0.305
#>    b2   -1.098      0.716 -1.534    0.125 -2.50  0.305
#>    b3    0.233      0.781  0.299    0.765 -1.30  1.764
#>    b4   -0.595      0.647 -0.919    0.358 -1.86  0.674
#>    b5   -0.418      0.647 -0.645    0.519 -1.69  0.851
#>    b6   -5.026      2.195 -2.290    0.022 -9.33 -0.725
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

f <- function(x) predict(x, type = "response", newdata = mtcars)
p <- hypotheses(mod, FUN = f)
head(p)
#> 
#>  Term Estimate Std. Error     z Pr(>|z|)   2.5 % 97.5 %
#>    b1  0.25005     0.1342 1.863  0.06243 -0.0130 0.5131
#>    b2  0.25005     0.1342 1.863  0.06243 -0.0130 0.5131
#>    b3  0.55803     0.1926 2.898  0.00376  0.1806 0.9355
#>    b4  0.35560     0.1483 2.399  0.01646  0.0650 0.6462
#>    b5  0.39710     0.1550 2.561  0.01043  0.0932 0.7010
#>    b6  0.00652     0.0142 0.459  0.64635 -0.0213 0.0344
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high 
#> 

# Equivalence, non-inferiority, and non-superiority tests
mod <- lm(mpg ~ hp + factor(gear), data = mtcars)
p <- predictions(mod, newdata = "median")
hypotheses(p, equivalence = c(17, 18))
#> 
#>  Estimate Std. Error    z Pr(>|z|) 2.5 % 97.5 %  hp gear p (NonInf) p (NonSup)
#>      19.7          1 19.6   <0.001  17.7   21.6 123    3    0.00404      0.951
#>  p (Equiv)
#>      0.951
#> 
#> Columns: rowid, estimate, std.error, statistic, p.value, conf.low, conf.high, mpg, hp, gear, statistic.noninf, statistic.nonsup, p.value.noninf, p.value.nonsup, p.value.equiv 
#> 

mfx <- avg_slopes(mod, variables = "hp")
hypotheses(mfx, equivalence = c(-.1, .1))
#> 
#>  Term Estimate Std. Error     z Pr(>|z|)   2.5 %  97.5 % p (NonInf) p (NonSup)
#>    hp  -0.0669      0.011 -6.05   <0.001 -0.0885 -0.0452    0.00135     <0.001
#>  p (Equiv)
#>    0.00135
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high, statistic.noninf, statistic.nonsup, p.value.noninf, p.value.nonsup, p.value.equiv 
#> 

cmp <- avg_comparisons(mod, variables = "gear", hypothesis = "pairwise")
hypotheses(cmp, equivalence = c(0, 10))
#> 
#>               Term Estimate Std. Error     z Pr(>|z|) 2.5 % 97.5 % p (NonInf)
#>  (4 - 3) - (5 - 3)    -3.94       2.05 -1.92   0.0543 -7.95 0.0727      0.973
#>  p (NonSup) p (Equiv)
#>      <0.001     0.973
#> 
#> Columns: term, estimate, std.error, statistic, p.value, conf.low, conf.high, statistic.noninf, statistic.nonsup, p.value.noninf, p.value.nonsup, p.value.equiv 
#>